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Coordination with Local Information
Munther A. Dahleh
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,[email protected]
Alireza Tahbaz-Salehi
Columbia Business School, Columbia University, New York, New York 10027, [email protected]
John N. Tsitsiklis
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,[email protected]
Spyros I. Zoumpoulis
Decision Sciences Area, INSEAD, 77300 Fontainebleau, France, [email protected]
We study the role of local information channels in enabling coordination among strategic agents. Building on the standardfinite-player global games framework, we show that the set of equilibria of a coordination game is highly sensitive tohow information is locally shared among different agents. In particular, we show that the coordination game has multipleequilibria if there exists a collection of agents such that (i) they do not share a common signal with any agent outsideof that collection and (ii) their information sets form an increasing sequence of nested sets. Our results thus extend theresults on the uniqueness and multiplicity of equilibria beyond the well-known cases in which agents have access to purelyprivate or public signals. We then provide a characterization of the set of equilibria as a function of the penetration of localinformation channels. We show that the set of equilibria shrinks as information becomes more decentralized.
Keywords : coordination; local information; social networks; global games.Subject classifications : games/group decisions: economics; network/graphs.Area of review : Special Issue on Information and Decisions in Social and Economic Networks.History : Received July 2013; revisions received October 2014, January 2015; accepted February 2015. Published online
in Articles in Advance November 12, 2015.
1. Introduction
Coordination problems lie at the heart of many economicand social phenomena such as bank runs, social uprisings,and the adoption of new standards or technologies. Thecommon feature of these phenomena is that the benefit oftaking a specific action to any given individual is highlysensitive to the extent to which other agents take the sameaction. The presence of such strong strategic complemen-tarities, coupled with the self-fulfilling nature of the agents’expectations, may then lead to coordination failures: indi-viduals may fail to take the action that is in their bestcollective interest.
Bank runs present a concrete (and classic) example ofthe types of coordination failures that may arise due tothe presence of strategic complementarities. In this context,any given depositor has strong incentives to withdraw hermoney from the bank if (and only if) she expects that alarge fraction of other depositors would do the same. Thus,a bank run may emerge, not because of any financial dis-tress at the bank, but rather as a result of the self-fulfillingnature of the depositors’ expectations about other deposi-tors’ behavior (Diamond and Dybvig 1983). Similarly, inthe context of adoption of new technologies with strongnetwork effects, consumers may collectively settle for an
inferior product simply because they expect other agents todo the same (Argenziano 2008).1
Given the central role of self-fulfilling beliefs in coor-dination games, it is natural to expect the emergence ofcoordination failures to be highly sensitive to the availabil-ity and distribution of information across different agents.In fact, since the seminal work of Carlsson and van Damme(1993), which initiated the global games literature, it hasbeen well known that the set of equilibria of a coordina-tion game depends on whether the information available tothe agents is public or private: the same game that exhibitsmultiple equilibria in the presence of public signals mayhave a unique equilibrium if the information were insteadonly privately available to the agents.The global games literature, however, has for the most
part only focused on the role of public and private informa-tion while ignoring the effects of local information chan-nels in facilitating coordination. This is despite the fact thatin many real world scenarios, local information channelsplay a key role in enabling agents to coordinate on differ-ent actions. For instance, it is by now conventional wis-dom that protesters in many recent antigovernment upris-ings throughout the Middle East used decentralized modesof communication (such as word-of-mouth communications
and Internet-based social media platforms) to coordinateon the time and location of street protests (Ali 2011).Similarly, adopters of new technologies that exhibit strongnetwork effects may rely on a wide array of informa-tion sources (such as blogs, professional magazines, expertopinions), not all of which are used by all other potentialadopters in the market.
Motivated by these observations, we study the role oflocal information channels in enabling coordination amongstrategic agents. Building on the standard finite-playerglobal games framework, we show that the set of equilibriaof a coordination game is highly sensitive to how infor-mation is locally shared among agents. More specifically,rather than restricting our attention to cases in which infor-mation is either purely public or private, we also allow forthe presence of local signals that are observable to subsetsof the agents. The presence of such local sources of infor-mation guarantees that some (but not necessarily all) infor-mation is common knowledge among a group of agents,with important implications for the determinacy of equi-libria. Our main contribution is to provide conditions foruniqueness and multiplicity of equilibria based solely onthe pattern of information sharing among agents. Our find-ings thus provide a characterization of the extent to whichcoordination failures may arise as a function of which pieceof information is available to each agent.
As our main result, we show that the coordination gamehas multiple equilibria if there exists a collection of agentssuch that (i) they do not share a common signal with anyagent outside of that collection and (ii) their informationsets form an increasing sequence of nested sets, which werefer to as a filtration. This result is a consequence of thefact that agents in such a collection face limited strate-gic uncertainty—that is, uncertainty concerning the equilib-rium actions—about one another,2 which transforms theircommon signals into a de facto coordination device. Tosee this in the most transparent way, consider two agentswhose information sets form a filtration. It is clear that theagent with the larger information set (say, agent 2) facesno uncertainty in predicting the equilibrium action of theagent with the smaller information set (agent 1). More-over, even though the latter is not informed about otherpotential signals observed by the former, the realizationsof such signals may not be extreme enough to push agent2 toward either action, thus making it optimal for her tosimply follow the behavior of agent 1. In other words, forcertain realizations of signals, the agent with the smallerinformation set becomes pivotal in determining the pay-off maximizing action of agent 2. Consequently, the set ofsignals that are common between the two can effectivelyfunction as a coordination device, leading to the emergenceof multiple equilibria.
We then focus on a special case of our model by assum-ing that each agent observes a single signal. We providean explicit characterization of the set of equilibria in termsof the commonality in agents’ information sets and show
that the set of equilibrium strategies enlarges if informationis more centralized. In other words, as the signals becomemore publicly available, the set of states under which agentscan coordinate on either action grows. This result is dueto the fact that more information concentration reduces theextent of strategic uncertainty among the agents, even if itdoes not impact the level of fundamental uncertainty.We then use our characterization results to study the set
of equilibria in large coordination games. We show that asthe number of agents grows, the game exhibits multipleequilibria if and only if a nontrivial fraction of the agentshave access to the same signal. Our result thus shows thatif the size of the subsets of agents with common knowledgeof a signal does not grow at the same rate as the number ofagents, the information structure is asymptotically isomor-phic to a setting in which all signals are private. Conse-quently, all agents face some strategic uncertainty regard-ing the behavior of almost every other agent, resulting in aunique equilibrium.Finally, even though our benchmark model assumes that
the agents’ information sets are common knowledge, wealso show that our results are robust to the introductionof small amounts of noise in the information structure ofthe game. In particular, by introducing uncertainty aboutthe information sets of different agents, we show that theequilibria of this new, perturbed game are close (in a formalsense) to the equilibria of the original game in which allinformation sets were common knowledge.In sum, our results establish that the distribution and
availability of information play a fundamental role in deter-mining coordination outcomes, thus highlighting the riskof abstracting from the intricate details of information dis-semination within coordination contexts. For example, onthe positive side, ignoring the distribution and reach of dif-ferent information sources can lead to wrong predictionsabout potential outcomes. Similarly, on the normative side(say for example, in the context of a central bank facing abanking crisis), the interventions of policymakers may turnout to be counterproductive if such actions are not informedby the dispersion and penetration of different informationchannels within the society.Related Literature. Our paper is part of the by now large
literature on global games. Initiated by the seminal workof Carlsson and van Damme (1993) and later expanded byMorris and Shin (1998, 2003), this literature focuses onhow the presence of strategic uncertainty may lead to theselection of a unique equilibrium in coordination games.The machinery of global games has since been used exten-sively to analyze various applications that exhibit an ele-ment of coordination, such as currency attacks (Morris andShin 1998), bank runs (Goldstein and Pauzner 2005), polit-ical protests (Edmond 2013), partnership investments (Das-gupta 2007), emergence of financial crises (Vives 2014),and adoption of technologies and products that exhibitstrong network effects (Argenziano 2008). For example,Argenziano (2008) uses the global games framework to
study a model of price competition in a duopoly with prod-uct differentiation and network effects. She shows that thepresence of a sufficient amount of noise in the public infor-mation available about the quality of the goods guaranteesthat the coordination game played by the consumers forany given set of prices has a unique equilibrium, and hencethe demand function for each product is well defined.
All the above papers, however, restrict their attention tothe case in which all information is either public or pri-vate. Our paper, on the other hand, focuses on more generalinformation structures by allowing for signals that are nei-ther completely public nor private and provides conditionsunder which the presence of such signals may lead to equi-librium multiplicity.
Our paper is also related to the works of Morris et al.(2015) and Mathevet (2014), who characterize the set ofequilibria of coordination games in terms of the agents’types while abstracting away from the information struc-ture of the game. Relatedly, Weinstein and Yildiz (2007)provide a critique of the global games approach by argu-ing that any rationalizable action in a game can alwaysbe uniquely selected by properly perturbing the agents’hierarchy of beliefs. Our work, on the other hand, pro-vides a characterization of the set of equilibrium strategieswhen the perturbation in the beliefs are generated by pub-lic, private, or local signals that are informative about theunderlying state. Despite being more restrictive in scope,our results shed light on the role of local information inenabling coordination.
A different set of papers studies how the endogeneity ofagents’ information structure in coordination games maylead to equilibrium multiplicity, thus qualifying the refine-ment of equilibria proposed by the standard global gamesargument. For example, Angeletos and Werning (2006),and Angeletos et al. (2006, 2007) show how prices, theaction of a policymaker, or past outcomes can functionas endogenous public signals that may restore equilibriummultiplicity in settings that would have otherwise exhib-ited a unique equilibrium. We study another natural settingin which agents may rely on overlapping (but not neces-sarily identical) sources of information and show how theinformation structure affects the outcomes of coordinationgames.
Our paper also belongs to the strand of literature thatfocuses on the role of local information channels and socialnetworks in shaping economic outcomes. Some recentexamples include Acemoglu et al. (2011), Golub and Jack-son (2010), Jadbabaie et al. (2012, 2013), and Galeotti et al.(2013). For example, Acemoglu et al. (2011) study howpatterns of local information exchange among few agentscan have first-order (and long-lasting) implications for theactions of others.
Within this context, however, our paper is more closelyrelated to the subset of papers that studies coordinationgames over networks, such as Galeotti et al. (2010) and
Chwe (2000). Galeotti et al. (2010) focus on strategic inter-actions over networks and characterize how agents’ inter-actions with their neighbors and the nature of the gameshape individual behavior and payoffs. They show that thepresence of incomplete information about the structure ofthe network may lead to the emergence of a unique equi-librium. The key distinction between their model and oursis in the information structure of the game and the natureof payoffs. Whereas they study an environment in whichindividuals care about the actions of their neighbors (whoseidentities they are uncertain about), we study a setting inwhich network effects are only reflected in the game’sinformation structure: individuals need to make deductionsabout an unknown parameter and the behavior of the restof the agents while relying on local sources of information.Chwe (2000), on the other hand, studies a coordination
game in which individuals can inform their neighbors oftheir willingness to participate in a collective risky behav-ior. Thus, our work shares two important aspects with thatof Chwe (2000): not only do both papers study games withstrategic complementarities, but also consider informationstructures in which information about the payoff-relevantparameters are locally shared among different individuals.Nevertheless, the two papers focus on different questions.Whereas Chwe’s main focus is on characterizing the set ofnetworks for which, regardless of their prior beliefs, agentscan coordinate on a specific action, our results provide acharacterization of how the penetration of different chan-nels of information can impact coordination outcomes.Outline of the Paper. The rest of the paper is organized
as follows: §2 introduces our model. In §3, we present aseries of simple examples that capture the intuition behindour results. Section 4 contains our results on the role oflocal information channels in the determinacy of equilibria.We then provide a characterization of the set of equilibriain §5 and show that the equilibrium set shrinks as informa-tion becomes more decentralized. Section 6 concludes. Allproofs are presented in the appendix.
2. Model
Our model is a finite-agent variant of the canonical modelof global games studied by Morris and Shin (2003).
2.1. Agents and Payoffs
Consider a coordination game played by n agents whose setwe denote by N = 81121 0 0 0 1n9. Each agent can take one oftwo possible actions, ai 2 80119, which we refer to as thesafe and risky actions, respectively. The payoff of taking thesafe action is normalized to zero, regardless of the actionsof other agents. The payoff of taking the risky action, onthe other hand, depends on (i) the number of other agentswho take the risky action and (ii) an underlying state ofthe world à 2✓, which we refer to as the fundamental. Inparticular, the payoff function of agent i is
where k = Pnj=1 aj is the number of agents who take the
risky action and è2 80111 0 0 0 1n9⇥✓!✓ is a function thatis Lipschitz continuous in its second argument. Throughout,we impose the following assumptions on the agents’ payofffunction, which are standard in the global games literature.
Assumption 1. Function è4k1 à5 is strictly increasing in kfor all à. Furthermore, there exists a constant ê> 0 such
that è4k1 à5Éè4kÉ 11 à5> ê for all à and all kæ 1.
The above assumption captures the presence of an ele-ment of coordination between agents. In particular, takingeither action becomes more attractive the more other agentstake that action. For example, in the context of a bankrun, each depositor’s incentive to withdraw her depositsfrom a bank not only depends on the solvency of the bank(i.e., its fundamental) but also increases with the numberof other depositors who decide to withdraw. Similarly, inthe context of adoption of technologies that exhibit net-work effects, each user finds adoption more attractive themore widely the product is adopted by other users in themarket. Thus, the above assumption simply guarantees thatthe game exhibits strategic complementarities. The secondpart of Assumption 1 is made for technical reasons andstates that the payoff of switching to the risky action whenone more agent takes action 1 is uniformly bounded frombelow, regardless of the value of à.
Assumption 2. Function è4k1 à5 is strictly decreasing
in à for all k.
That is, any given individual has less incentive to takethe risky action if the fundamental takes a higher value.Thus, taking other agents’ actions as given, each agent’soptimal action is decreasing in the state. For instance, in thebank run context, if the fundamental value à corresponds tothe financial health of the bank, the depositors’ withdrawalincentives are stronger the closer the bank is to insolvency.Similarly, in the context of the adoption of network goods,à can represent the quality of the status quo technologyrelative to the new alternative in the market: regardless ofthe market shares, a higher à makes adoption of the newtechnology less attractive. Finally, we impose the followingassumption:
Assumption 3. There exist constants à1 à̄ 2 ✓ satisfying
à< à̄ such that
(i) è4k1 à5> 0 for all k and all à< à.
(ii) è4k1 à5< 0 for all k and all à> à̄.
Thus, each agent strictly prefers to take the safe (risky)action for sufficiently high (low) states of the world, irre-spective of the actions of other agents. If, on the other hand,the underlying state belongs to the so-called critical region6à1 à̄7, then the optimal behavior of each agent depends onher beliefs about the actions of other agents. Thus, onceagain, in the bank run context, the above assumption sim-ply means that each depositor finds withdrawing (leaving)
her deposit a dominant action if the bank is sufficientlydistressed (healthy).In summary, agents face a coordination game with strong
strategic complementarities in which the value of coordi-nating on the risky action depends on the underlying stateof the world. Furthermore, particular values of the statemake either action strictly dominant for all agents.
2.2. Information and Signals
As in the canonical global games model, agents are notaware of the realization of the fundamental. Rather, theyhold a common prior belief on à 2 ✓, which for simplic-ity we assume to be the (improper) uniform distributionover the real line. Furthermore, conditional on the realiza-tion of à, a collection 4x11 0 0 0 1xm5 2 ✓m of noisy signalsis generated, where xr = à+ ér . We assume that the noiseterms 4é11 0 0 0 1 ém5 are independent from à and are drawnfrom a continuous joint probability density function withfull support over ✓m.Not all agents, however, can observe all realized sig-
nals. Rather, agent i has access to a nonempty subset Ii ✓8x11 0 0 0 1xm9 of the signals, which we refer to as her infor-mation set.3 This assumption essentially captures that eachagent may have access to multiple sources of informationabout the underlying state. Going back to the bank runexample, this means that each depositor may obtain someinformation about the health of the bank via a variety ofinformation sources (such as the news media, the results ofstress tests released by the regulatory agencies, inside infor-mation, expert opinions, etc.). The depositor would thenuse the various pieces of information available to her todecide whether to run on the bank or not.The fact that agent i’s information set can be any
(nonempty) subset of 8x11 0 0 0 1xm9 means that the extentto which any given signal xr is observed may vary acrosssignals. For example, if xr 2 Ii for all i, then xr is essen-tially a public signal observed by all agents. On the otherhand, if xr 2 Ii for some i but xr 62 Ij for all j 6= i, then xris a private signal of agent i. Any signal that is neitherprivate nor public can be interpreted as a local source ofinformation observed only by a proper subset of the agents.The potential presence of such local signals is our point ofdeparture from the canonical global games literature, whichonly focuses on games with purely private or public sig-nals. Note that in either case, following the realization ofsignals, the mean of agent i’s posterior belief about the fun-damental is simply equal to her Bayesian estimate ⇧6à ó Ii7.We remark that even though agents are uncertain about
the realizations of the signals they do not observe, theinformation structure of the game—that is, the collectionof information sets 8Ii9i2N—is common knowledge amongthem.4 Therefore, a pure strategy of agent i is simply amapping si2 ✓
óIi ó ! 80119, where óIió denotes the cardinal-ity of agent i’s information set.Finally, we impose the following mild technical assump-
tion, ensuring that the agents’ payoff function is integrable.
Assumption 4. For any collection of signals 8xr9r2H , H ✓811 0 0 0 1m9 and all k 2 801 0 0 0 1n9,
⇧6óè4k1 à5ó ó 8xr9r2H 7<à0
We end this section by remarking that given that the pay-off functions have nondecreasing differences in the actionsand the state, the underlying game is a Bayesian game withstrategic complementarities. Therefore, by Van Zandt andVives (2007, Theorem 14), the game’s Bayesian Nash equi-libria form a lattice whose extremal elements are monotonein types. This lattice structure also implies that the extremalequilibria are symmetric, in the sense that each agent’s strat-egy only depends on the signals she observes but not on heridentity (Vives 2008). Therefore, to characterize the rangeof equilibria, without loss of generality, we can restrict ourattention to symmetric equilibria in threshold strategies.
3. A Simple Example
Before proceeding to our general results, we present a sim-ple example and show how the presence of local signalsdetermines the set of equilibria. Consider a game consistingof n = 3 agents and, for expositional simplicity, supposethat
è4k1 à5= kÉ 1nÉ 1
É à0 (1)
It is easy to verify that this payoff function, which is sim-ilar to the one in the canonical finite-player global gamesliterature such as Morris and Shin (2000, 2003), satisfiesAssumptions 1–3 with à= 0 and à̄= 1. We consider threedifferent information structures, contrasting the cases inwhich agents have access to only public or private informa-tion to a case with a local signal. Throughout this section,we assume that the noise terms ér are mutually indepen-dent and normally distributed with mean zero and varianceë2 > 0.
Public Information. First, consider a case in which allagents observe the same public signal x, that is, Ii = 8x9for all i 2 8112139. Thus, no agent has any private informa-tion about the state. It is easy to verify that under such aninformation structure, the coordination game has multipleBayesian Nash equilibria. In particular, for any í 2 60117,the strategy profile in which all agents take the risky actionif and only if x < í is an equilibrium, regardless of thevalue of ë . Consequently, as the public signal becomesinfinitely accurate (i.e., as ë ! 0), the underlying gamehas multiple equilibria as long as the underlying state àbelongs to the critical region 60117.
Private Information. Next, consider the case in which allagents have access to a different private signal. In particu-lar, suppose that three signals 4x11x21x35 2✓3 are realizedand that xi is privately observed by agent i; that is, Ii = 8xi9for all i. As is well known from the global games litera-ture, the coordination game with privately observed signals
has an essentially unique Bayesian Nash equilibrium. Toverify that the equilibrium of the game is indeed unique,it is sufficient to focus on the set of equilibria in thresholdstrategies, according to which each agent takes the riskyaction if and only if her private signal is smaller than agiven threshold.5 In particular, let íi denote the thresholdcorresponding to the strategy of agent i. Taking the strate-gies of agents j and k as given, agent i’s expected pay-off of taking the risky action is equal to ⇧6è4k1 à5 ó xi7=12 6⇣4xj < íj ó xi5 + ⇣4xk < ík ó xi57 É xi. For íi to corre-spond to an equilibrium strategy of agent i, she has to beindifferent between taking the safe and the risky actionswhen xi = íi. Hence, the collection of thresholds 4í11 í21 í35corresponds to a Bayesian Nash equilibrium of the game ifand only if for all permutations of i, j , and k, we have
íi =12Í
✓íj É íi
ëp2
◆+ 1
2Í
✓ík É íi
ëp2
◆1
where Í4 · 5 denotes the cumulative distribution function ofthe standard normal. Note that à ó xi ⇠ N 4xi1ë
25, and asa result, xj ó xi ⇠ N 4xi12ë25. It is then immediate to ver-ify that í1 = í2 = í3 = 1/2 is the unique solution of theabove system of equations. Thus, in the (essentially) uniqueequilibrium of the game, agent i takes the risky action ifshe observes xi < 1/2, whereas she takes the safe actionif xi > 1/2. Following standard arguments from Carlssonand van Damme (1993) or Morris and Shin (2003), onecan show that this strategy profile is also the (essentially)unique strategy profile that survives the iterated eliminationof strictly dominated strategies. Consequently, in contrastto the game with public information, as ë ! 0, all agentschoose the risky action if and only if à < 1/2. This obser-vation shows that in the limit as signals become arbitrarilyprecise and all the fundamental uncertainty is removed, thepresence of strategic uncertainty among the agents leads tothe selection of a unique equilibrium.Local Information. Finally, consider the case where only
two signals 4x11x25 are realized and the agents’ informa-tion sets are I1 = 8x19 and I2 = I3 = 8x29; that is, agent 1observes a private signal whereas agents 2 and 3 haveaccess to the same local source of information. All theinformation available to agents 2 and 3 is common knowl-edge between them, which distinguishes this case from thecanonical global game model with private signals.To determine the extent of equilibrium multiplicity, once
again it is sufficient to focus on the set of equilibria inthreshold strategies. Let í1 and í2 = í3 denote the thresh-olds corresponding to the strategies of agents 1, 2, and 3,respectively. If agent 1 takes the risky action, she obtains anexpected payoff of ⇧6è4k1 à5 ó x17 = ⇣4x2 < í2 ó x15É x1.On the other hand, the expected payoff of taking the riskyaction to agent 2 (and, by symmetry, agent 3) is given by
⇧6è4k1 à5 ó x27= 12 6⇣4x1 < í1 ó x25+ ⌧8x2 < í297É x21
where ⌧8 · 9 denotes the indicator function. Thus, for thresh-olds í1 and í2 to correspond to a Bayesian Nash equilib-rium, it must be the case that
í1 =Í
✓í2 É í1
ëp2
◆0 (2)
Furthermore, agents 2 and 3 should not have an incentiveto deviate, which requires that their expected payoffs haveto be positive for any x2 < í2 and negative for any x2 > í2,leading to the following condition:
2í2 É 1∂Í
✓í1 É í2
ëp2
◆∂ 2í20 (3)
It is easy to verify that the pair of thresholds 4í11 í25 simul-taneously satisfying (2) and (3) is not unique. In particular,as ë ! 0, for every í1 2 61/312/37, there exists some í2close enough to í1 such that the profile of threshold strate-gies 4í11 í21 í25 is a Bayesian Nash equilibrium. Conse-quently, as the signals become very precise, the underlyinggame has multiple equilibria as long as à 2 6 13 1
23 7.
Thus, even though there are no public signals, the pres-ence of common knowledge between a proper subset of theagents restores the multiplicity of equilibria. In other words,the local information available to agents 2 and 3 serves as acoordination device, enabling them to predict one another’sactions. The presence of strong strategic complementaritiesin turn implies that agent 1 will use his private signal asa predictor of how agents 2 and 3 coordinate their actions.Nevertheless, because of the presence of some strategicuncertainty between agents 2 and 3 on the one hand andagent 1 on the other, the set of rationalizable strategies isstrictly smaller compared to the case where all three agentsobserve a public signal. Therefore, the local signal (par-tially) refines the set of equilibria of the coordination game,though not to the extent that would lead to uniqueness.
4. Local Information and Equilibrium
Multiplicity
The examples in the previous section show that the set ofequilibria in the presence of local signals may not coincidewith the set of equilibria under purely private or publicsignals. In this section, we provide a characterization ofthe role of local information channels in determining theequilibria of the coordination game presented in §2. Beforepresenting our main result, we define the following concept:
Definition 1. The information sets of agents in C =8i11 0 0 0 1 ic9 form a filtration if Ii1 ✓ Ii2 ✓ · · ·✓ Iic .
Thus, the information sets of agents in set C constitute afiltration if they form a nested sequence of increasing sets.This immediately implies that the signals of the agent withthe smallest information set is common knowledge amongall agents in C. We have the following result:
Theorem 1. Suppose that Assumptions 1–4 are satisfied.
Also, suppose that there exists a subset of agents C ✓ Nsuch that
(a) The information sets of agents in C form a filtration.
(b) Ii \ Ij =ô for any i 2C and j 62C.Then the coordination game has multiple Bayesian Nash
equilibria.
The above result thus shows that, in general, the pres-ence of a cascade of increasingly rich observations guaran-tees equilibrium multiplicity. Such filtration provides somedegree of common knowledge among the subset of agentsin C, reduces the extent of strategic uncertainty they faceregarding each other’s behavior, and as a result leads to theemergence of multiple equilibria. Therefore, in this sense,Theorem 1 generalizes the standard, well-known results inthe global games literature to the case when agents haveaccess to local information channels that are neither publicnor private.We remark that even though the presence of local signals
may lead to equilibrium multiplicity, the set of BayesianNash equilibria does not necessarily coincide with that of agame with purely public signals. Rather, as we show in thenext section, the set of equilibria crucially depends on thenumber of agents in C as well as the information structureof other agents.To see the intuition underlying Theorem 1, suppose that
the information sets of agents in some set C form a fil-tration. It is immediate that there exists an agent i 2 Cwhose signals are observable to all other agents in C. Con-sequently, agents in C face no uncertainty (strategic or oth-erwise) in predicting the equilibrium actions of agent i.Furthermore, the signals in Ij\Ii provide agent j 2 C\8i9with information about the underlying state à beyond whatagent i has access to. Nevertheless, there exist realizationsof such signals such that the expected payoff of taking therisky action to any agent j 2 C\8i9 would be positive ifand only if agent i takes the risky action. In other words,for such realizations of signals, agent i’s action becomespivotal in determining the payoff maximizing actions of allother agents in C. Consequently, signals in Ii can essen-tially serve as a coordination device among C’s members,leading to multiple equilibria.Finally, note that condition (b) of Theorem 1 plays a cru-
cial role in the above argument. In particular, it guaranteesthat agents in C effectively face an induced coordinationgame among themselves, in which they can use the signalsin Ii as a coordination device.
4.1. The Role of Information Filtrations
In this subsection, we use a series of examples to exhibitthe main intuition underlying Theorem 1 and highlight therole of its different assumptions.
Example 1. Consider a two-player game with linear pay-offs è4k1 à5 = k É 1É à as in (1). Furthermore, suppose
that the realized signals 4x11x25 are normally distributed,with agents’ information sets given by
I1 = 8x191 I2 = 8x11x290
It is immediate that the above information structure satisfiesthe assumptions of Theorem 1.
As before, focusing on threshold strategies is sufficient fordetermining the set of equilibria. Let í1 denote the thresh-old corresponding to agent 1’s strategy. The fact that x1 is asignal that is observable to both agents implies that agent 2can predict the equilibrium action of agent 1 with certaintyand change her behavior accordingly. On the other hand, theexpected payoff of agent 2 when she takes the risky actionis given by a1 É ⇧6à ó x11x27= a1 É 4x1 + x25/2. These twoobservations together imply that the best response of agent 2would be a threshold strategy 4í21 í 0
25 constructed as follows:if x1 < í1, then agent 2 takes the risky action if and only if4x1 + x25/2< í2, whereas if x1 æ í1, agent 2 takes the riskyaction if and only if 4x1 + x25/2< í 0
2.Therefore, the expected payoff of taking the risky action
to agent 2 is equal to
⇧6è4k1 à5 ó x11x27= ⌧8x1 < í19É 12 4x1 + x250
The fact that agent 2 has to be indifferent between takingthe safe and the risky actions at her two thresholds impliesthat í2 = 1 and í 0
2 = 0. On the other hand, the expectedpayoff of taking the risky action to agent 1 is given by
⇧6è4k1 à5 ó x17= ⇣4x2 < 2í2 É x1 ó x15⌧8x1 < í19
+⇣4x2 < 2í 02 É x1 ó x25⌧8x1 æ í19É x10
Given that í1 captures the threshold strategy of agent 1, theabove expression has to be positive if and only if x1 < í1,or equivalently:
Í
✓É í1
ëp2
◆∂ í1 ∂Í
✓2É í1
ëp2
◆0
It is immediate to verify that the value of í1 that satisfiesthe above inequalities is not unique, thus guaranteeing themultiplicity of equilibria.
To interpret this result, note that whenever ⇧6à ó x11x27 240115, agent 2 finds it optimal to simply follow the behav-ior of agent 1, regardless of the realization of signal x2.This is because (i) within this range, agent 2’s belief aboutthe underlying state à lies inside the critical region; and(ii) agent 2 can perfectly predict the action of agent 1, mak-ing agent 1 pivotal and hence leading to multiple equilibria.
It is important to note that the arguments following The-orem 1 and the above example break down if agent 1 hasaccess to signals that are not in the information set ofagent 2, even though there are some signals that are com-mon knowledge between the two agents. To illustrate this
point, consider a three-agent variant of the above example,where the agents’ information sets are given by
I1 = 8x21x391 I2 = 8x31x191 I3 = 8x11x290 (4)
Thus, even though there is one signal that is commonknowledge among any given pair of agents, the informationsets of no subset of agents form a filtration. We have thefollowing result:
Proposition 2. Suppose that agents’ information sets are
given by (4). There exists ë̄ such that if ë > ë̄ , then the
game has an essentially unique equilibrium.
Thus, the coordination game has a unique equilibriumdespite the fact that any pair of agents shares a commonsignal. This is because, unlike Theorem 1 and Example 1above, every agent faces some strategic uncertainty aboutall other agents in the game, hence guaranteeing that nocollection of signals can serve as a coordination deviceamong a subset of agents.In addition to the presence of a subset of agents C whose
information sets form a filtration, Theorem 1 also requiresthat the information set of no agent outside C contain anyof the signals observable to agents in C. To clarify the roleof this assumption in generating multiple equilibria, onceagain consider the above three-player game but instead sup-pose that agents’ information sets are given by
I1 = 8x191 I2 = 8x11x291 I3 = 8x290 (5)
It is immediate to verify that the conditions of Theorem 1are not satisfied, even though there exists a collection ofagents whose information sets form a filtration. We havethe following result:
Proposition 3. Suppose that agents’ information sets are
given by (5). Then the game has an essentially unique equi-
librium. Furthermore, as ë ! 0, all agents choose the riskyaction if and only if à< 1/2.
Thus, as fundamental uncertainty is removed, informa-tion structure (5) induces the same (essentially) uniqueequilibrium as the case in which all signals are private.This is despite the fact that the information sets of agentsin C = 81129 form a filtration.To understand the intuition behind this result, it is
instructive to compare the above game with the gamedescribed in Example 1. Notice that in both games, agentswith the larger information sets do not face any uncer-tainty (strategic or otherwise) about predicting the actionsof agents with smaller information sets. Nevertheless, theabove game exhibits a unique equilibrium, whereas thegame in Example 1 has multiple equilibria. The key dis-tinction lies in the different roles played by the extra piecesof information available to the agents with larger informa-tion sets. Recall that in Example 1, there exists a set ofrealizations of signals for which agent 2 finds it optimal to
imitate the action of agent 1. Therefore, under such condi-tions, agent 2 uses x2 solely as a means of obtaining a moreprecise estimate of the underlying state à. This, however,is in sharp contrast with what happens under informationstructure (5). In this case, signal x2 plays a second role inthe information structure of agent 2: it not only providesan extra piece of information about à but also serves asa perfect predictor of agent 3’s equilibrium action. Thus,even if observing x2 does not change the mean of agent 2’sposterior belief about à by much, it still provides her withinformation about the action that agent 3 is expected to takein equilibrium. This extra piece of information, however,is not available to agent 1. Thus, even as ë ! 0, agents 1and 3 face some strategic uncertainty regarding the equilib-rium action of one another and, as a consequence, regardingthat of agent 2. The presence of such strategic uncertaintiesimplies that the game with information structure (5) wouldexhibit a unique equilibrium.
We conclude our argument by showing that even thoughthe conditions of Theorem 1 are sufficient for equilibriummultiplicity, they are not necessary. To see this, once againconsider a variant of the game in Example 1 but insteadassume that there are four agents whose information setsare given by
Note that the above information structure does not satisfythe conditions of Theorem 1. Nevertheless, one can showthat the game has multiple equilibria:
Proposition 4. Suppose that agents’ information sets are
given by (6). Then the game has multiple Bayesian Nash
equilibria.
The importance of the above result is highlighted whencontrasted with the information structure (5) and Proposi-tion 3. Note that in both cases, agent 2 has access to thesignals available to all other agents. However, in informa-tion structure (6), signal x2 is also available to an additionalagent, namely, agent 4. The presence of such an agentwith an information set identical to that of agent 3 meansthat agents 3 and 4 face no strategic uncertainty regardingone another’s actions. Therefore, even though they may beuncertain about the equilibrium actions of agents 1 and 2,there are certain realizations of signal x2 under which theyfind it optimal to imitate one another’s action, thus leadingto the emergence of multiple equilibria.
5. Local Information and the Extent of
Multiplicity
Our analysis thus far was focused on the dichotomybetween multiplicity and uniqueness of equilibria. How-ever, as the example in §3 shows, even when the gameexhibits multiple equilibria, the set of equilibria depends onhow information is locally shared between different agents.
In this section, we provide a characterization of the set ofall Bayesian Nash equilibria as a function of the informa-tion sets of different agents. Our characterization quantifiesthe dependence of the set of rationalizable strategies on theextent to which agents observe common signals.To explicitly characterize the set of equilibria, we restrict
our attention to a game with linear payoff functions givenby (1). We also assume that m ∂ n signals, denoted by4x11 0 0 0 1xm5 2 ✓m, are realized, where the noise terms4é11 0 0 0 1 ém5 are mutually independent and normally dis-tributed with mean zero and variance ë2 > 0. Furthermore,we assume that each agent observes only one of the real-ized signals; that is, for any given agent i, her informationset is Ii = 8xr9 for some 1 ∂ r ∂ m. Finally, we denotethe fraction of agents that observe signal xr by cr , and letc= 4c11 0 0 0 1cm5. Since each agent observes a single signal,we have c1 + · · · + cm = 1. Note that as in our benchmarkmodel in §2, we assume that the allocation of signals toagents is deterministic, prespecified, and common knowl-edge. Our next result provides a simple characterization ofthe set of all rationalizable strategies as ë ! 0.
Theorem 5. Let si denote a threshold strategy of agent i.As ë ! 0, strategy si is rationalizable if and only if
si4x5=(1 if x < í
0 if x > í̄1
where
í = 1É í̄ = n
24nÉ 1541Éòcò225∂
12
and òcò2 denotes the Euclidean norm of vector c.
The above theorem shows that the distance between thethresholds of the “largest” and “smallest” rationalizablestrategies depends on how information is locally sharedbetween different agents. More specifically, a smaller òcò2implies that the set of rationalizable strategies wouldshrink.Note that òcò2 is essentially a proxy for the extent to
which agents observe common signals: it takes a smallervalue whenever any given signal is observed by feweragents. Hence, a smaller value of òcò2 implies that agentswould face higher strategic uncertainty about one another’sactions, even when all the fundamental uncertainty isremoved as ë ! 0. As a consequence, the set of ratio-nalizable strategies shrinks as the Euclidean norm of cdecreases. In the extreme case that agents’ information isonly in the form of private signals (that is, when m= n andcr = 1/n for all r), the upper and lower thresholds coin-cide 4í = í̄ = 1/25, implying that the equilibrium strategiesare essentially unique. This is indeed the case that corre-sponds to maximal level of strategic uncertainty. On theother hand, when all agents observe the same public sig-nal (i.e., when m = 1 and òcò2 = 1), they face no strate-gic uncertainty about each other’s actions, and hence all
undominated strategies are rationalizable. Thus, to summa-rize, the above two extreme cases coincide with the stan-dard results in the global games literature. Theorem 5 abovegeneralizes those results by characterizing the equilibria ofthe game in the intermediate cases in which signals areneither fully public nor private.
Recall that since the Bayesian game under considerationis monotone supermodular in the sense of Van Zandt andVives (2007), there exist a greatest and a smallest BayesianNash equilibrium, both of which are in threshold strate-gies. Moreover, by Milgrom and Roberts (1990), all profilesof rationalizable strategies are “sandwiched” between thesetwo equilibria. Therefore, Theorem 5 also provides a char-acterization of the set of equilibria of the game, showingthat a higher level of common knowledge, captured via alarger value for òcò2, implies a larger set of equilibria. Notethat if m<n, by construction, there are at least two agentswith identical information sets. Theorem 5 then implies thatí < 1/2< í̄ , which means that the game exhibits multipleequilibria, an observation consistent with Theorem 1.
A simple corollary to Theorem 5 implies that with m<nsources of information, the set of Bayesian Nash equilibriais largest when mÉ 1 agents each observe a private signaland nÉm+ 1 agents have access to the remaining signal.In this case, common knowledge of signals among sucha large group of agents minimizes the extent of strategicuncertainty and hence leads to the largest set of equilibria.On the other hand, the set of equilibria shrinks wheneverthe sizes of the sets of agents with access to the samesignal are more equalized. In particular, the case in whichcr = 1/m for all r corresponds to the highest level of inter-group strategic uncertainty, leading to the greatest extent ofrefinement of rationalizable strategies.
5.1. Large Coordination Games
Recall from Theorem 1 that the existence of two agents iand j with identical information sets is sufficient to guar-antee equilibrium multiplicity, irrespective of the numberof agents in the game or how much other agents care aboutcoordinating with i and j . In particular, no matter howinsignificant and uninformed the two agents are, the merefact that i and j face no uncertainty regarding each other’sbehavior leads to equilibrium multiplicity. On the otherhand, as Theorem 5 and the preceding discussion show,even under information structures that lead to multiplicity,the set of equilibria still depends on the extent to whichagents observe common signals. To further clarify the roleof local information in determining the size of the equilib-rium set, we next study large coordination games.
Formally, consider a sequence of games 8G4n59n2� pa-rametrized by the number of agents, in which each agent ican observe a single signal, and assume that the noise termsin the signals are mutually independent and normally dis-tributed with mean zero and variance ë2 > 0. We have thefollowing corollary to Theorem 5.
Proposition 6. The sequence of games 8G4n59n2� exhibits
an (essentially) unique equilibrium asymptotically as
n!à and in the limit as ë ! 0 if and only if the size of
the largest set of agents with a common observation grows
sublinearly in n.
Thus, as the number of agents grows, the game exhibitsmultiple equilibria if and only if a nontrivial fraction of theagents have access to the same signal. Even though such asignal is not public—in the sense that it is not observed byall agents—the fact that it is common knowledge amonga nonzero fraction of the agents implies that it can func-tion as a powerful enough coordination device and henceinduce multiple equilibria. On the other hand, if the sizeof the largest subset of agents with common knowledge ofa signal does not grow at the same rate as the number ofagents, information is diverse and effectively private: anyagent faces strategic uncertainty regarding the behavior ofmost other agents, even as all the fundamental uncertaintyis removed (ë ! 0). Consequently, as the number of agentsgrows, the set of equilibrium strategies of each agent col-lapses to a single strategy.
5.2. Robustness to Uncertainty in the
Information Structure
So far, we have assumed that the information structure ofthe game is common knowledge. In this subsection, werelax this assumption and show that our results are robustto a small amount of uncertainty regarding the informationstructure of the game.To capture this idea formally, consider a game G⇤ with
information structure Q⇤ = 8I11 0 0 0 1 In9 that is commonknowledge among the agents, where each signal is observedby more than a single agent. Furthermore, consider a per-turbed version of G⇤ in which the information structure isdrawn according to a probability distribution å that is inde-pendent from the realization of à and the signals. In partic-ular, the likelihood that a certain information structure Q isrealized is given by å4Q5.6 This means that as long as åis not degenerate, agents are not only uncertain about therealization of the underlying state à and other agents’ sig-nals, but they may also have incomplete information aboutone another’s information sets.We consider the case where the cardinality of any agent’s
information set is identical across all possible informationstructures for which å4Q5> 0. This assumption guaranteesthat any strategy of a given agent is a mapping from thesame subspace of ✓n to 80119 regardless of the realizationof the information structure, thus allowing us to compareagents’ strategies for different probability measures å. Wehave the following result:7
Theorem 7. Consider a sequence of probability distribu-
tions 8ås9às=1 such that lims!àås4Q
⇤5 = 1, and let Gs
denote the game whose information structure is drawn
according to ås . Then for almost all equilibria of G⇤, there
exists a large enough s̄ such that for all s > s̄, that equilib-rium is also a Bayesian Nash equilibrium of Gs .
The above result thus highlights that even if the informa-tion structure of the game is not common knowledge amongthe agents, the equilibria of the perturbed game would beclose to the equilibria of the original game. Consequently,it establishes that our earlier results on the multiplicity ofequilibria are also robust to the introduction of a smallamount of uncertainty about the information structure ofthe game.
As a last remark, we note that in the above result, wedid not allow agents to obtain extra, side signals about therealization of the information structure of the game. Never-theless, a similar argument shows that the set of equilibriais robust to small amounts of noise in the information struc-ture of the game, even if agents obtain such informativesignals.
6. Conclusions
Many social and economic phenomena (such as bank runs,adoption of new technologies, social uprisings, etc.) exhibitan element of coordination. In this paper, we focused onhow the presence of local information channels can affectthe likelihood and possibility of coordination failures insuch contexts. In particular, by introducing local signals—signals that are neither purely public nor private—to thecanonical global game models studied in the literature, weshowed that the set of equilibria depends on how informa-tion is locally shared among agents. Our results establishthat the coordination game exhibits multiple equilibria ifthe information sets of a group of agents form an increas-ing sequence of nested sets: the presence of such a filtrationremoves agents’ strategic uncertainty about one another’sactions and leads to multiple equilibria.
We also provided a characterization of how the extentof equilibrium multiplicity is determined by the extent towhich subsets of agents have access to common informa-tion. In particular, we showed that the size of the equilib-rium set is increasing in the standard deviation of the frac-tions of agents with access to the same signal. Our resultthus shows that the set of equilibria shrinks as informationbecomes more decentralized in the society.
On the theoretical side, our results highlight the impor-tance of local information channels in global games byunderscoring how the introduction of a signal commonlyobserved by even a small number of agents may lead toequilibrium multiplicity in an environment that would haveotherwise exhibited a unique equilibrium. On the moreapplied side, our results show that incorporating local com-munication channels between agents may be of first-orderimportance in understanding many phenomena that exhibitan element of coordination. In particular, they highlight therisk of abstracting from the intricate details of informationdissemination within such contexts. For instance, policyinterventions that are not informed by the dispersion andpenetration of different information channels among agentsmay turn out to be counterproductive.
Acknowledgments
The authors are grateful to the editors and two anonymous refer-ees for helpful feedback and suggestions. They also thank DaronAcemoglu, Marios Angeletos, Antonio Penta, Ali Shourideh, andparticipants at the Workshop on the Economics of Networks, Sys-tems, and Computation and the joint Workshop on Pricing andIncentives in Networks and Systems, where preliminary versionsof this work were presented (Dahleh et al. 2011, 2013); the Inter-disciplinary Workshop on Information and Decision in SocialNetworks; and the Cambridge Area Economics and ComputationDay. This research was partially supported by the Air Force Officeof Scientific Research, Grant FA9550-09-1-0420.
Appendix. Proofs
Notation and Preliminary Lemmas
We first introduce some notation and prove some preliminary lem-mas. Recall that a pure strategy of agent i is a mapping si2 ✓
óIi ó !80119, where Ii denotes i’s information set. Thus, a pure strategysi can equivalently be represented by the set Ai ✓✓óIi ó over whichagent i takes the risky action; i.e.,
Ai = 8yi 2✓óIi ó2 si4yi5= 191
where yi = 4xr5r2Ii denotes the collection of the realized signals inthe information set of agent i. Hence, a strategy profile can equiv-alently be represented by a collection of sets A = 4A11 0 0 0 1An5
over which agents take the risky action. We denote the set of allstrategies of agent i by Ai and the set of all strategy profiles by A.Given the strategies of other agents, AÉi, we denote the expectedpayoff to agent i of the risky action when she observes yi byVi4AÉi ó yi5. Thus, a best response mapping BRi2 AÉi ! Ai isnaturally defined as
BRi4AÉi5= 8yi 2✓óIi ó2 Vi4AÉi ó yi5> 090 (7)
Finally, we define the mapping BR2 A!A as the product of thebest response mappings of all agents; that is,
BR4A5=BR14AÉ15⇥ · · ·⇥BRn4AÉn50 (8)
The BR4 · 5 mapping is monotone and continuous. More formally,we have the following lemmas:
Lemma 1 (Monotonicity). Consider two strategy profiles A
and A0such that A✓A0
. (We write A✓A0whenever Ai ✓A0
i for
all i.) Then BR4A5✓BR4A05.
Proof. Fix an agent i and consider yi 2BRi4AÉi5, which by defi-nition satisfies Vi4AÉi ó yi5> 0. Because of the presence of strate-gic complementarities (Assumption 1), we have Vi4A
0Éi ó yi5 æ
Vi4AÉi ó yi5 and, as a result, yi 2BRi4A0Éi5. É
Lemma 2 (Continuity). Consider a sequence of strategy pro-
Proof. Clearly, Ak ✓ Aà and by Lemma 1, BR4Ak5✓ BR4Aà5for all k. Thus,
à[
k=1
BR4Ak5✓BR4Aà50
To prove the reverse inclusion, suppose that yi 2BRi4AàÉi5, which
implies that Vi4AàÉi ó yi5 > 0. On the other hand, for any à and
any observation profile 4y11 0 0 0 1yn5, we have
limk!à
ui4ai1 skÉi4yÉi51 à5= ui4ai1 s
àÉi4yÉi51 à51
where sk and sà are strategy profiles corresponding to sets Ak andAà, respectively. Thus, by the dominated convergence theorem,
limk!à
Vi4AkÉi ó yi5= Vi4A
àÉi ó yi51 (9)
where we have used Assumption 4. Therefore, there exists r 2� large enough such that Vi4A
rÉi ó yi5 > 0, implying that yi 2Sà
k=1 BR4Aki 5. This completes the proof. É
Throughout the rest of the proofs, we let 8Rk9k2� denote thesequence of strategy profiles defined recursively as
R1 =ô1
Rk+1 =BR4Rk50 (10)
Thus, any strategy profile A that survives k rounds of iteratedelimination of strictly dominated strategies must satisfy Rk+1 ✓A.Consequently, A survives the iterated elimination of strictly dom-inated strategies only if R✓A, where
R=à[
k=1
Rk0 (11)
Proof of Theorem 1
Without loss of generality, let C = 811 0 0 0 1 `9 and I1 ✓ I2 · · ·✓ I`,where ` æ 2. Before proving the theorem, we first define an“induced coordination game” between agents in C = 811 0 0 0 1 `9 asfollows: suppose that agents in C play the game described in §2,except the actions of agents outside of C are prescribed by thestrategy profile R defined in (10) and (11). More specifically, thepayoff of taking the risky action to agent i 2C is given by
è̃4k1 à5=è
✓k+
X
j 62C⌧8yj 2Rj91 à
◆1
where k=P`j=1 aj denotes the number of agents in C who take
the risky action. As in the benchmark game in §2, the payoff oftaking the safe action is normalized to zero.
For this game, define the sequence of strategy profiles8R0k9k2� as
R01 =ô1 (12)
R0k+1 =BR4R0k51 (13)
and set R0 = Sàk=1R
0k. Clearly, the strategy profile 4R011 0 0 0 1R
0`5
corresponds to a Bayesian Nash equilibrium of the induced gamedefined above. Furthermore, it is immediate that R0
i = Ri for alli 2C.
We have the following lemma:
Lemma 3. The induced coordination game between the ` agents
in C has an equilibrium strategy profile B = 4B11 0 0 0 1B`5 such
that ã✓óIi ó4Bi \ R0i5 > 0 for all i 2 C, where ã✓r is the Lebesgue
measure in ✓rand the equilibrium strategy profile R0
is given by
(12) and (13).
Proof. We denote yj = 8xr 2 xr 2 Ij9. Recall that any strategyprofile of agents in C can be represented as a collection of setsA = 4A11 0 0 0 1A`5, where Aj is the set of realizations of yj overwhich agent j takes the risky action. On the other hand, recallthat since I1 ✓ I2 · · ·✓ I`, agent j observes the realizations of sig-nals yi for all i ∂ j . Therefore, any strategy Aj of agent j canbe recursively captured by indexing it to the realizations of sig-nals 4y11 0 0 0 1yiÉ15. More formally, given the realization of signals4y11 0 0 0 1y`5 and the strategies 4A11 0 0 0 1AjÉ15, agent j takes therisky action if and only if
yj 2A4s110001sjÉ15
j 1
where
si = ⌧8yi 2A4s110001siÉ15i 9
for all i ∂ j É 1. In other words, j first considers the set of sig-nals 4y11 0 0 0 1yjÉ15 and, based on their realizations, takes the riskyaction if yj 2 A
4s110001sjÉ15
j . Note that in the above notation, ratherthan being captured by a single set Aj ✓✓óIj ó, the strategy of agentj is captured by 2jÉ1 different sets of the form A
4s110001sjÉ15
j . Thus,the collection of sets
A= 8A4s110001sjÉ15
j 91∂j∂`1 s280119`É1 (14)
capture the strategy profile of the agents 811 0 0 0 1 `9 in the inducedcoordination game. Finally, to simplify notation, we let sj =4s11 0 0 0 1 sj5 and Sj =
PjÉ1i=1 si.
With the above notation at hand, we now proceed with theproof of the lemma. Note that since the induced coordinationgame is a game of strategic complementarities, it has at least oneequilibrium. In fact, the iterated elimination of strictly dominatedstrategies in (12) and (13) leads to one such equilibrium A= R0.Let A, represented in (14), denote one such strategy profile, wherewe assume that agents take the safe action whenever they areindifferent between the two actions.
For the strategy profile A to be an equilibrium, the expectedpayoff of paying the risky action to agent j has to be positivefor all yj 2 A
s110001sjÉ1j , and nonpositive for yj 62 A
s110001sjÉ1j . In other
words, if yj 2As110001sjÉ1j , then
X
411sj+110001srÉ15
⇧6è̃41+S`1à5 óyj 7⇣4y`2As`1yk2A4skÉ15
k 8sk=11
yk 62A4skÉ15k 8sk=02 j <k<` óyj5
+X
411sj+110001srÉ15
⇧6è̃4S`1à5 óyj 7⇣4y` 62As`1yk2A4skÉ15
k 8sk=11
yk 62A4skÉ15k 8sk=02 j <k<` óyj5>03
whereas, on the other hand, if yj 62As110001sjÉ1j , then
To understand the above expressions, note that in equilibrium,agent j faces no strategic uncertainty regarding the behavior ofagents i∂ jÉ1. However, to compute her expected payoff, agent jneeds to condition on different realizations of signals 4sj+110001s`5.Furthermore, she needs to take into account that the realizationsof such signals not only affect the belief of agents k>j about theunderlying state à but also determine which set As4kÉ15
k they woulduse for taking the risky action.
Given the equilibrium strategy profile A whose properties wejust studied, we now construct a new strategy profile D as fol-lows. Start with agent ` at the top of the chain and define a setD
4s110001s`É15` to be such that it satisfies
⇧6è̃41+S`1à5 óy`7>0 if y`2D4s110001s`É15` 1
⇧6è̃41+S`1à5 óy`7∂0 if y`yD4s110001s`É15` 0
Note that for any given s, the set Ds` is essentially an extension of
As` to the whole space ✓óI`ó in the sense that the two sets coincide
with one another over the subset of signals realizations
8y`2 yi2AsiÉ1
i if si=1 and yiyAsiÉ1
i if si=090
Similarly, (and recursively) define the sets Dj as extensions ofthe sets Aj such that they satisfy the following properties: if yj 2D
4s110001sjÉ15
j , thenX
411sj+110001srÉ15
⇧6è̃41+S`1à5 óyj 7⇣4y`2Ds`1yk2D4skÉ15
k 8sk=11
yk 62D4skÉ15k 8sk=03k>j óyj5
+X
411sj+110001srÉ15
⇧6è̃4S`1à5 óyj 7⇣4y` 62Ds`1yk2D4skÉ15
k 8sk=11
yk 62D4skÉ15k 8sk=03k>j óyj5>03 (15)
and, on the other hand, if yj 62Ds110001sjÉ1j , then
X
401sj+110001srÉ15
⇧6è̃41+S`1à5 óyj 7⇣4y`2Ds`1yk2D4skÉ15
k 8sk=11
yk 62D4skÉ15k 8sk=03k>j óyj5
+X
401sj+110001srÉ15
⇧6è̃4S`1à5 óyj 7⇣4y` 62Ds`1yk2D4skÉ15
k 8sk=11
yk 62D4skÉ15k 8sk=03k>j óyj5∂00 (16)
Note that given that the sets D are simply extensions of sets A,by construction, the strategy profile 4D110001D`5 also forms aBayesian Nash equilibrium. In fact, it essentially is an alternativerepresentation of the equilibrium strategy profile A.
Now, consider the expected payoff to agent 1, at the bottom ofthe chain. In equilibrium, agent 1 should prefer to take the riskyaction in set D1. In other words,
f 4y15>0 8y12D1 (17)
f 04y15∂0 8y1 62D11 (18)
where
f 4y15=X
s280119`É12s1=1
⇧6è̃41+S`1à5 óy17⇣4y`2Ds`1
yk2D4skÉ15k 8sk=11yk 62A4skÉ15
k 8sk=02 1<k<` óy15+
X
s280119`É12s1=1
⇧6è̃4S`1à5 óy17⇣4y` 62Ds`1
yk2D4skÉ15k 8sk=11yk 62A4skÉ15
k 8sk=02 1<k<` óy151
and
f 04y15=X
s280119`É1 2s1=0
⇧6è̃41+S`1à5 óy17⇣4y`2Ds`1
yk2D4skÉ15k 8sk=11yk 62A4skÉ15
k 8sk=02 k>1 óy15+
X
s280119`É1 2s1=0
⇧6è̃4S`1à5 óy17⇣4y` 62Ds`1
yk2D4skÉ15k 8sk=11yk 62A4skÉ15
k 8sk=02 k>1 óy151
are simply determined by evaluating inequalities (15) and (16) foragent 1. Notice that unlike the definition of f 4y15, in the definitionof f 04y15 we have to start from s1=0 because agent 1 is notsupposed to take the risky action whenever y1 62D1.
We have the following lemma, the proof of which is providedlater on:
Lemma 4. Let X=8y12 f 4y15>09 and Y =8y12 f04y15>09. Then
there exists a set B1 such that Y ✓B1✓X and ã✓óI1 ó4B1„D15>0,where ã✓óI1 ó refers to the Lebesgue measure and „ denotes the
symmetric difference.
In view of the above lemma, we now use the set B1 to constructa new strategy profile B represented as the following collectionsof sets:
B=4B118D4s110001sjÉ15
j 92∂j∂`1s280119`É150
In other words, agent 1 takes the risky action if and only if y12B1.The rest of the agents follow the same strategies as prescribedby strategy profile D except for the fact that rather than choos-ing their D sets based on whether y1 belongs to D1 or not, theychoose the sets based on whether y12B1 or not. By construc-tion, the strategy profile above is an equilibrium. Note that sinceY ✓B1✓X, agent 1 is best responding to the strategies of all otheragents. Furthermore, given that for j 6=1 we have that B4sjÉ15
j =D
4sjÉ15j , inequalities (15) and (16) are also satisfied, implying that
agent j also best responds to the behavior of all other agents,hence, completing the proof. É
We now present the proof of Theorem 1.Proof of Theorem 1. Lemma 3 establishes that the induced
coordination game between agents in C has an equilibriumB=4B110001B`5 that is distinct from the equilibrium R0 derivedvia iterated elimination of strictly dominated strategies in (12)and (13). We now use this strategy profile to construct an equi-librium for the original n-agent game.
Define the strategy profile B̃=4B110001B`1R`+1 0001Rn5 for then-agent game, where Rj is defined in (10) and (11). Lemma 3immediately implies that B̃✓BR4B̃5. Define the sequence of strat-egy profiles 8Hk9k2� as
H 1= B̃
Hk+1=BR4Hk50
Given that H1✓H 2, Lemma 1 implies that Hk✓Hk+1 for all k.Thus, H=Sà
k=1Hk is well defined and by continuity of the BR
operator (Lemma 2) satisfies H=BR4H5. As a consequence, His also a Bayesian Nash equilibrium of the game that, in lightof Lemma 3, is distinct from R. Note that R is itself a BayesianNash equilibrium of the game because it is a fixed point of thebest response operator. É
Proof of Lemma 4. The definitions of sets X and Y imme-diately imply that Y ✓D1✓X. On the other hand, the secondpart of Assumption 1 implies that ã✓óI1 ó4X\Y 5>0. Consequently,there exists a set B1 distinct from D1 such that Y ✓B✓X andã✓óI1 ó4B1„D15>0. This completes the proof. É
Proof of Proposition 2
Recall the sequence of strategy profiles Rk and its limit R definedin (10) and (11), respectively. By Lemma 2, R=BR4R5, whichimplies that yi2Ri if and only if Vi4RÉi óyi5>0. We have thefollowing lemma.
Lemma 5. There exists a strictly decreasing function h2 ✓!✓such that Vi4RÉi óyi5=0 if and only if xj =h4xl5, where yi=4xj 1xl5 and i, j , and l are different.
Proof. Using an inductive argument, we first prove that for all k(i) Vi4R
kÉi óyi5 is continuously differentiable in yi,
(ii) Vi4RkÉi óyi5 is strictly decreasing in both arguments,
4xj 1xl5=yi, and(iii) ó°Vi4R
kÉi óyi5/°xj ó2 61/21Q7,
where j 6= i and
Q= ëp3è+1
2ëp3èÉ2
0
The above clearly hold for k=1 because Vi4ôóyi5=É4xj+xl5/2.Now suppose that (i)–(iii) are satisfied for some kæ1. By theimplicit function theorem,8 there exists a continuously differen-tiable function hk2 ✓!✓ such that
Vi4RkÉi óxj 1hk4xj55=01
and É2Q∂h0k4xj5∂É1/42Q5. (Given the symmetry between the
three agents, we drop the agent’s index for function hk.) Themonotonicity of hk implies that Vi4R
kÉi óyi5>0 if and only if xj <
hk4xl5. Therefore,
Vi4Rk+1Éi óyi5
= 12 6⇣4yj 2Rk+1
j óyi5+⇣4yl2Rk+1l óyi57É 1
2 4xj+xl5
= 12 6⇣4xi <hk4xl5 óyi5+⇣4xi <hk4xj5 óyi57É 1
2 4xj+xl5
= 12
Í
✓hk4xl5É4xj+xl5/2
ëp3/2
◆
+Í
✓hk4xj5É4xj+xl5/2
ëp3/2
◆�É 124xj+xl51
which immediately implies that Vi4Rk+1Éi óyi5 is continuously differ-
entiable and strictly decreasing in both arguments. Furthermore,°
°xjVi4R
k+1Éi óyi5
=É 1
2ëp6î
✓hk4xl5É4xj+xl5/2
ëp3/2
◆
+ 2h0k4xj5É1
2ëp6
î
✓hk4xj5É4xj+xl5/2
ëp3/2
◆É 121
which guarantees
É12É 1+2Q
2ëp3è
∂ °
°xjVi4R
k+1Éi óyi5∂É1
21
completing the inductive argument because12+ 1+2Q
2ëp3è
=Q0
Now using (9) and the implicit function theorem once again com-pletes the proof. É
Setting Vi4RÉi óyi5=0 and any solution yi=4xj 1xl) ofVi4RÉi óyi5=0 satisfies h4xl5=xj and h4xj5=xl imply thatxj+xl=1. Therefore,
Ri=�4xj 1xl52✓22 1
2 4xj+xl5<12
0
Hence, in any strategy profile that survives iterated elimination ofstrictly dominated strategies, an agent takes the risky action when-ever the average of the two signals she observes is less than 1/2.A symmetrical argument implies that in any strategy profile thatsurvives the iterated elimination of strictly dominated strategies,the agent takes the safe action whenever the average of her signalsis greater than 1/2. Thus, the game has an essentially unique ratio-nalizable strategy profile and hence an essentially unique BayesianNash equilibrium. É
Proof of Proposition 3
It is sufficient to show that there exists an essentially unique equi-librium in monotone strategies. Note that because of symmetry,the strategies of agents 1 and 3 in the extremal equilibria of thegame are identical. Denote the (common) equilibrium thresholdof agents 1 and 3’s threshold strategies by í 2 60117, in the sensethat they take the risky action if and only if their observation isless than í . Then the expected payoff of taking the risky actionto agent 2 is equal to
⇧6è4k1à5 óx11x27= 12 6⌧8x1<í9+⌧8x2<í9É4x1+x2570
First suppose that í>1/2. Then the best response of agent 2 isto take the risky action if either (i) x1+x2∂1 or (ii) x11x2∂í
hold. On the other hand, for í to correspond to the threshold of anequilibrium strategy of agent 1, her expected payoff of taking therisky action has to be positive whenever x1<í . In other words,
12 ⇣4x2<í óx15+ 1
2 6⇣4x1+x2∂1 óx15+⇣41Éx1∂x2∂í óx157æx1
for all x1<í . As a result, for any x1<í , we have
Taking the limit of the both sides of the above inequality asx1"í implies that í∂1/2. This, however, contradicts the originalassumption that í>1/2. A similar argument would also rule outthe case that í<1/2. Hence, í corresponds to the threshold ofan equilibrium strategy of agents 1 and 3 only if í=1/2. As aconsequence, in the essentially unique equilibrium of the game,agent 2 takes the risky action if and only if x1+x2<1. This provesthe first part of the proposition. The proof of the second part isimmediate. É
Proof of Proposition 4
One again, we can simply focus on the set of equilibria in thresh-old strategies. Let í1, í3, and í4 denote the thresholds of agents 1,3, and 4, respectively. Note that by symmetry, í3=í4. Also letfunction f2 denote the strategy of agent 2, in the sense that agent 2takes the risky action if and only if x2<f24x15.
The expected value of taking the risky action to agent 1 isgiven by
⇧6è4k1à5 óx17= 13 ⇣4x2<f24x15 óx15+ 2
3 ⇣4x2<í3 óx15Éx10
Thus, for í1 to be the equilibrium threshold strategy for agent 1,the above expression has to be zero at x1=í1:
í1=13Í
✓f24í15Éí1
ëp2
◆+ 23Í
✓í3Éí1
ëp2
◆(19)
On the other hand, the expected value of taking the risky actionto agents 3 and 4 is given by
⇧6è4k1à5 óx27= 13 ⇣4x1<í1 óx25+ 1
3 ⇣4x1<f É12 4x25 óx25
+ 13⌧8x2<í39Éx21
where f É12 is the inverse function of f2. For í3 to be the equi-
librium threshold strategy for agents 3 and 4, the above expres-sion has to be positive for x2<í3 and negative for x2>í3, whichimplies
í2É13∂ 1
3Í
✓í1Éí3
ëp2
◆+ 13Í
✓f É12 4í35Éí3
ëp2
◆∂í20 (20)
It is easy to verify that the triplet 4í11f21í35 that satisfies condi-tions (19) and (20) simultaneously is not unique. É
Proof of Theorem 5
As already mentioned, the Bayesian game under considerationis monotone supermodular in the sense of Van Zandt and Vives(2007), which ensures that the set of equilibria has well-definedmaximal and minimal elements, each of which is in thresholdstrategies. Moreover, by Milgrom and Roberts (1990), all profilesof rationalizable strategies are “sandwiched” between these twoequilibria. Hence, to characterize the set of rationalizable strate-gies, it suffices to focus on threshold strategies and determine thesmallest and largest thresholds that correspond to Bayesian Nashequilibria of the game.
Denote the threshold corresponding to the strategy of an agentwho observes signal xr with ír . The profile of threshold strategiescorresponding to thresholds 8ír9 is a Bayesian Nash equilibriumof the game if and only if for all r ,
n
nÉ1
X
j 6=r
cjÍ
✓íjÉxr
ëp2
◆Éxr ∂0 for xr >ír
ncrÉ1nÉ1
+ n
nÉ1
X
j 6=r
cjÍ
✓íjÉxr
ëp2
◆Éxr æ0 for xr <ír 1
where the first (second) inequality guarantees that the agent hasno incentive to deviate to the risky (safe) action when the sig-nal she observes is above (below) threshold ír . Taking the limitas xr converges to ír from above in the first inequality impliesthat in any symmetric, Bayesian Nash equilibrium of the game inthreshold strategies,
4nÉ15íænHc1
where í= 6í110001ím7 is the vector of thresholds and H 2✓m⇥m isa matrix with zero diagonal entries, and off-diagonal entries givenby Hjr =Í44íjÉír5/ë
p25. Therefore,
24nÉ15c0íænc04H 0+H5c
=nc041105cÉnc0c1
where 1 is the vector of all ones and Í4z5+Í4Éz5=1. Conse-quently,
24nÉ15c0íæn41Éòcò2250
Finally, given that óírÉíj ó!0 as ë!0 and òcò1=1, the left-hand side of the above inequality converges to 24nÉ15í⇤ forsome constant í⇤; as a result, the smallest possible threshold thatcorresponds to a Bayesian Nash equilibrium is equal to
í= n
24nÉ1541Éòcò2250
The expression for í̄ is derived analogously. É
Proof of Proposition 6
We first prove that if the size of the largest set of agents with acommon observation grows sublinearly in n, then asymptoticallyas n!à, the game has a unique equilibrium. We denote thevector of the fractions of agents observing each signal by c4n5 tomake explicit the dependence on the number of agents n. Note thatif the size of the largest set of agents with a common observationgrows sublinearly in n, then
limn!à
òc4n5òà=01
where òzòà is the maximum element of vector z. On the otherhand, by Hölder’s inequality,
òc4n5ò22∂òc4n5ò1 ·òc4n5òà0
Given that the elements of vector c4n5 add up to one,limn!àòc4n5ò2=0. Consequently, Theorem 5 implies that thresh-olds í4n5 and í̄4n5 characterizing the set of rationalizable strate-gies of the game of size n satisfy
Thus, asymptotically, the game has an essentially unique BayesianNash equilibrium.
To prove the converse, suppose that the size of the largestset of agents with a common observation grows linearly in n,which means that òc4n5òà remains bounded away from zero asn!à. Furthermore, the inequality òc4n5òà∂òc4n5ò2 immedi-ately implies that òc4n5ò2 also remains bounded away from zeroas n!à. Hence, by Theorem 5,
limsupn!à
í4n5<1/2
liminfn!à
í̄4n5>1/21
guaranteeing asymptotic multiplicity of equilibria as n!à. É
Proof of Theorem 7
Pick a nonextremal equilibrium in the game with informationstructure Q⇤=8I110001In9 and denote it by 4A110001An5, accordingto which agent i takes the risky action if and only if yi2Ai. Wedenote the expected payoff of taking the risky action to agent iunder information structure Q⇤ by V ⇤
i 4AÉi óyi5. Given that eachsignal is observed by at least two agents and that the strategy pro-file 4A110001An5 is a nonextremal equilibrium, there exists Ö>0such that
V ⇤i 4AÉi óyi5>Ö for all yi2Ai
V ⇤i 4AÉi óyi5<ÉÖ for all yi 62Ai0
Now consider the sequence of games 8Gs9 in which the informa-tion structure is drawn according to the sequence of probabilitydistributions 8ås9. It is easy to see that in the game indexed bys, agent i’s expected payoff of taking the risky action, when allother agents follow strategies AÉi, is given by
V4s5i 4AÉi óyi5=
X
Q
ås4Q5V Qi 4AÉi óyi51
where we denote the expected payoff of taking the risky actionto agent i under information structure Q by V Q
i 4AÉi óyi5. Sincelims!àås4Q
⇤5=1, it is immediate that the above expression con-verges to V ⇤
i 4AÉi óyi5.Consequently, given that V ⇤
i 4AÉi óyi5 is uniformly boundedaway from zero for all yi, there exists a large enough s̄ such thatfor s>s̄, the expected payoff of taking the risky action in gameGs when other agents follow strategies AÉi is strictly positive ifyi2Ai, whereas it would be strictly negative if yi 62Ai. Thus, forany s>s̄, the strategy profile 4A110001An5 is also a Bayesian Nashequilibrium of game Gs . É
Endnotes
1. See Morris and Shin (2003) for more examples.2. Note that strategic uncertainty is distinct from fundamentaluncertainty. Whereas fundamental uncertainty simply refers to theagents’ uncertainty about the underlying, payoff-relevant state ofthe world, strategic uncertainty refers to their uncertainty concern-ing the equilibrium actions of one another.3. With some abuse of notation, we use Ii to denote both theset of actual signals observed by agent i as well as the set ofindices of the signals observed by that agent. Zoumpoulis (2014)discusses the special case where the agents are organized in anetwork, such that each agent observes a signal that pertains toher and the signals of her neighbors.
4. We relax this assumption in §5.2.5. As already mentioned, this is a consequence of the underlyinggame being a Bayesian game with strategic complementarities,and hence the extremal equilibria are monotone in types. For adetailed study of Bayesian games with strategic complementari-ties, see Van Zandt and Vives (2007).6. Note that since the information set of any given agent is a sub-set of the finitely many realized signals, å is a discrete probabilitymeasure with finite support.7. We thank an anonymous referee for suggesting this result.8. See, for example, Hadamard’s global implicit function theoremin Krantz and Parks (2002).
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Munther A. Dahleh is the William A. Coolidge Professor ofElectrical Engineering and Computer Science at MIT. He is cur-rently the acting director of the Engineering Systems Division
and Director-designate of a new organization addressing majorsocietal problems through unification of the intellectual pillarsin Statistics, Information and Decision Systems, and Human andInstitution Behavior.
Alireza Tahbaz-Salehi is Daniel W. Stanton Associate Pro-fessor of Business at Columbia Business School. His researchfocuses on the implications of network economies for businesscycle fluctuations and financial stability.
John N. Tsitsiklis is the Clarence J. Lebel Professor of Elec-trical Engineering and Computer Science at MIT. He chairs theCouncil of the Harokopio University in Athens, Greece. Hisresearch interests are in systems, optimization, communications,control, and operations research. He has been a recipient of anOutstanding Paper Award from the IEEE Control Systems Society(1986), the M.I.T. Edgerton Faculty Achievement Award (1989),the Bodossakis Foundation Prize (1995), an ACM SigmetricsBest Paper Award (2013), and a co-recipient of two INFORMSComputing Society prizes (1997, 2012). He is a member of theNational Academy of Engineering. In 2008, he was conferred thetitle of Doctor honoris causa, from the Université Catholique deLouvain.
Spyros I. Zoumpoulis is an assistant professor of DecisionSciences at INSEAD. He is broadly interested in problems at theinterface of networks, learning with data, information, and deci-sion making.
CORRECTION
In the print version of this article, “Coordination with Local Information” by Munther A. Dahleh, Alireza Tahbaz-Salehi, John N. Tsitsiklis, and Spyros I. Zoumpoulis (Operations Research, Vol. 64, No. 3, pp. 622–637), the researcharticle was mislabeled as a technical note. “Technical Note” has been removed from the title in the online version andan erratum will be printed in the July–August issue.
